A064326 Generalized Catalan numbers C(-4; n).

1, 1, -3, 25, -251, 2817, -33843, 425769, -5537835, 73865617, -1004862179, 13888533561, -194475377243, 2752994728225, -39333541106835, 566464908534345, -8214515461250955, 119845125957958065, -1757855400878129475, 25906894146115000665, -383443906519878272955

Offset: 0

Author

Wolfdieter Lang, Sep 21 2001

Keywords

• sign
• easy

Comments

See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..830

Crossrefs

Cf. A064334.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9 +Sqrt(1+16*x))/(2*(5-x)) )); // G. C. Greubel, May 03 2019

Mathematica

CoefficientList[Series[(9 +Sqrt[1+16*x])/(2*(5-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)

PARI

my(x='x+O('x^30)); Vec((9 +sqrt(1+16*x))/(2*(5-x))) \\ G. C. Greubel, May 03 2019

Sage

def a(n):
    if n==0: return 1
    return hypergeometric([1-n, n], [-n], -4).simplify()
[a(n) for n in range(20)] # Peter Luschny, Nov 30 2014

Sage

((9 +sqrt(1+16*x))/(2*(5-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019

Formula

a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-4)^m/n.

a(n) = (1/5)^n*(1 + 4*Sum_{k=0..n-1} C(k)*(-4*5)^k), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).

G.f.: (1+4*x*c(-4*x)/5)/(1-x/5) = 1/(1-x*c(-4*x)) with c(x) g.f. of Catalan numbers A000108.

a(n) = hypergeometric([1-n, n], [-n], -4) for n > 0. - Peter Luschny, Nov 30 2014